The image shows a mathematical problem in Malay, asking about the quadratic function $f(x) = -x^2 + 6x - 5$. A straight line AB is parallel to the x-axis. We are required to determine the following:

(a) Koordinat titik A (Coordinates of point A)

Since point A is on the x-axis, the y-coordinate will be 0. To find the x-coordinate of A, we solve $f(x) = 0$.

$f(x) = -x^2 + 6x - 5 = 0$

Solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = -1$, $b = 6$, and $c = -5$. Substituting these values:

$x = \frac{-6 \pm \sqrt{6^2 - 4(-1)(-5)}}{2(-1)} = \frac{-6 \pm \sqrt{36 - 20}}{-2} = \frac{-6 \pm \sqrt{16}}{-2} = \frac{-6 \pm 4}{-2}$

Thus,

$x = \frac{-6 + 4}{-2} = 1 \quad \text{or} \quad x = \frac{-6 - 4}{-2} = 5$

So, point A is at $(1, 0)$.

(b) Persamaan paksi simetri (Equation of the axis of symmetry)

The axis of symmetry for a quadratic function $f(x) = ax^2 + bx + c$ is given by:

$x = \frac{-b}{2a}$

Substitute $a = -1$ and $b = 6$:

$x = \frac{-6}{2(-1)} = 3$

Thus, the axis of symmetry is $x = 3$.

(c) Koordinat titik B (Coordinates of point B)

Since point B is on the same horizontal line (parallel to the x-axis) as point A, and both points share the same y-coordinate, we know $B = (5, 0)$.

(d) Koordinat titik maksimum P (Coordinates of the maximum point P)

The maximum point (vertex) of the parabola is located at the axis of symmetry, $x = 3$. To find the y-coordinate, substitute $x = 3$ into the function $f(x)$:

$f(3) = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4$

Thus, the maximum point is $P = (3, 4)$.

Summary of Answers:

• (a) Koordinat titik A: $(1, 0)$
• (b) Persamaan paksi simetri: $x = 3$
• (c) Koordinat titik B: $(5, 0)$
• (d) Koordinat titik maksimum P: $(3, 4)$

Would you like further details or have any questions?

Here are 5 related questions to enhance your understanding:

1. How does the quadratic formula solve for roots of a quadratic equation?
2. What is the geometric meaning of the axis of symmetry for a quadratic graph?
3. Can you derive the vertex formula from completing the square method?
4. How do you determine if a parabola opens upwards or downwards?
5. Why do points A and B have the same y-coordinate if the line AB is parallel to the x-axis?

Tip: The vertex of a parabola is a critical point that helps determine the maximum or minimum value of a quadratic function.